The Open Mapping and Closed Graph Theorems in Topological Vector Spaces

THE main purpose of writing this monograph is to give a picture of the progress made in recent years in understanding three of the deepest results of Functional Analysis-namely, the open-mapping and closed­ graph theorems, and the so-called Krein-~mulian theorem. In order to facilitate the reading of this book, some of the important notions and well-known results about topological and vector spaces have been collected in Chapter 1. The proofs of these results are omitted for the reason that they are easily available in any standard book on topology and vector spaces e.g. Bourbaki [2], Keiley [18], or Köthe [22]. The results of Chapter 2 are supposed to be weil known for a study of topological vector spaces as weil. Most of the definitions and notations of Chapter 2 are taken from Bourbaki’s books [3] and [4] with some trimming and pruning here and there. Keeping the purpose of this book in mind, the presentation of the material is effected to give a quick resume of the results and the ideas very commonly used in this field, sacrificing the generality of some theorems for which one may consult other books, e.g. [3], [4], and [22]. From Chapter 3 onward, a detailed study of the open-mapping and closed-graph theorems as weil as the Krein-~mulian theorem has been carried out. For the arrangement of the contents of Chapters 3 to 7, see the Historical Notes (Chapter 8).